smooth quasigroups and loops: forty-ﬁve years of...

Comment.Math.Univ.Carolin. 41,2 (2000)377–400 377

Smooth quasigroups and loops:

forty-five years of incredible growth

Lev V. Sabinin

Abstract. The remarkable development of the theory of smooth quasigroups is surveyed.

Keywords: smooth quasigroups and loops, odules, loopuscular and odular algebras,Bol loops, Moufang loops, Bol algebras, Mal’cev algebras, nonlinear geometric algebra,nonassociative geometry, F-quasigroups, transsymmetric spaces, hyperalgebra, hypore-ductivity, pseudoreductivity

Classification: 20N05, 53A60, 53C22

0. There might be different opinions on the date of birth of the theory of smoothquasigroups. But nobody would deny that the pioneering work of Mal’cev[A.I. Mal’cev 55] was a milestone of exceptional importance in the area. Thuswe regard this very work as a starting point of smooth quasigroups and loopstheory. In this survey we concentrate our efforts only on main ideological achieve-ments; it would be not productive and, moreover, impossible to survey the wholescope of published papers in the field.Instead we present a large (but not comprehensive) bibliography on the subject.Near 1955, after remarkable results of the algebraic theory of quasigroups, the

spiritual state of World Mathematical Mind reached a proper strength to createsomething new. A very natural idea to generalize the Lie groups theory in anon-associative way obtained the shape and force and appeared in the paper[A.I. Mal’cev 55]. This initiated attempts to construct the foundation of smoothquasigroups and loops in different parts of the world (Russia, Japan, Germany,Central Europe).Some attempts of geometric nature were related to web geometry. We will not

be concerned with this matter, which is treated in the article of M. Akivis andV. Goldberg. Here we note only that web geometry gives a useful instrument tostudy isotopically invariant properties of quasigroups. The study of propertieswhich are not isotopically invariant needs different approaches.From [L.V. Sabinin 98a] one may see how, by means of the construction of

antiproduct, web theory is reduced to standard considerations of loop theory.Among those who influenced the development of the theory of smooth quasi-

groups, I would especially mention the following personalities (in alphabeticalorder): A.S. Fedenko, A.N. Grishkov, M. Kikkawa, O. Kowalski, E.N. Kuz’min,A.I. Ledger, O. Loos, A.I. Mal’cev, P. Nagy, L.V. Sabinin, A.A. Sagle, K. Stram-bach.

378 L.V. Sabinin

1. In the paper [A.I. Mal’cev 55] a local diassociative analytic loop Q = 〈Q, · , ε〉was studied. Diassociativity means that any two elements generate a subgroup.Since the multiplication in a loop is a binary operation and the loop is diassocia-tive, one may write the analogue of the Campbell-Hausdorff series, which dependsonly on one skew-symmetric bilinear operation (multiplication ×) in the tangentspace V = TεQ, at ε ∈ Q. Thus, at least locally, a diassociative loopQ = 〈Q, · , ε〉can be restored by means of its tangent algebra 〈V,+, 0V ,×〉. This algebra is abinary-Lie algebra, that is, any two of its elements generate a subalgebra whichis a Lie algebra. Any binary-Lie algebra generates, by means of the Campbell-Hausdorff formula, a diassociative local loop. Thus we get a diassociative smoothloops — binary-Lie algebras theory generalizing the Lie groups — Lie algebrastheory. Later, Gainov [A.T. Gainov 57] described a binary Lie algebra V by theidentities:

(1)

{

J(ξ, η, ξ × η) = 0, ξ × ξ = 0, ξ, η ∈ V,

J(ξ, η, ζ) = ξ × (η × ζ) + ζ × (ξ × η) + η × (ζ × ξ).

In the same paper of A.I. Mal’cev a smooth Moufang loop was considered, as well.A Moufang loop can be defined as a loop 〈Q, · , ε〉 with the identities

(2)Lx ◦ Ly ◦ Lx = Lx·(y·x), Lxz

def= x · z

(left Bol identity),

(3)Rx ◦ Ry ◦ Rx = R(x·y)·x, Rxz

def= z · x

(right Bol identity).

Being diassociative, a smooth Moufang loop can be infinitesimally described by askew-symmetric algebra with identities. Mal’cev called it a Moufang-Lie algebra.Now it is called a Mal’cev algebra. Its defining identities are:

(4) ξ × ξ = 0 , J(ξ, τ, , ξ × η) = J(ξ, τ, η)× ξ , ξ, η, τ ∈ V.

At that time the question whether any Mal’cev algebra uniquely defines a smoothMoufang loop or not was open. This question was answered in positive by Kuz’min[E.N. Kuz’min 70, 71]. Thus the infinitesimal theory for smooth Moufang loops— Mal’cev algebras has been constructed in the full analogy with the Lie groups— Lie algebras theory. Kuz’min classified, also, all simple Mal’cev algebras overR. With a few rather interesting exceptions, these turned out to be Lie algebras[E.N. Kuz’min 68]. Earlier the same, in fact, was done by Sagle [A.A. Sagle62a], who intensively studied Mal’cev algebras [A.A. Sagle 61, 62a, b]. Later someresults on global smooth Moufang loops were obtained [F.S. Kerdman 79].In the light of the above, we may speak of a Mal’cev school in the smooth

loops theory (Mal’cev-Gainov-Kuz’min and others), the approach of which was

Smooth quasigroups and loops: forty-five years of incredible growth 379

purely algebraic, without any concern for the geometry. Much later Grishkov[A.N. Grishkov 86a, b] finalized, to a certain extent, the above theory. He con-structed the structure theory of binary-Lie algebras. In particular, he showedthat any simple binary-Lie algebra (over R) is a Mal’cev algebra.There are still some open problems in the field: is the group generated by

left (right) translations Lx, Lxy = x · y , (Rx, Rxy = y · x) of a diassociativeC3-smooth loop a Lie group? Note that this is true for Moufang loops.

2. The other sources of the smooth quasigroup theory had their backgrounds indifferential geometry. O. Loos [O. Loos 69] retold the theory of symmetric spaces(see, for example, [S. Helgason 62, 78]) in the language of smooth quasigroups,although he did not use the term ‘quasigroup’ [O. Loos 69]. It was a remarkablebreakthrough, since earlier no good geometric examples of smooth quasigroupshad been discovered. It was followed by the notion of generalized symmetric spaces[A.J. Ledger 67], [A.J. Ledger, M. Obata 68], [A.S. Fedenko 73, 77], [O. Kowalski74, 80] which has been well studied. Again, despite the geometric approach, theseresults have a smooth quasigroups nature. In the language of smooth quasigroups,a generalized symmetric space is simply a smooth (partial or global) quasigroup〈Q, · 〉 with the identities

x · (x · y) = y (key identity),(5)

x · (y · z) = (x · y) · (x · z) (left distributive identity).(6)

3. Meanwhile, a very important fact was discovered by Kikkawa [M. Kikkawa 64].By means of parallel translations of geodesic arcs along geodesic arcs in an affinelyconnected manifold (M,∇), he constructed so-called geodesic loops of an affinelyconnected manifold. More precisely, M may be considered to have the binaryoperations (local or global)

(7) x ·a

ydef= Expx τa

x Exp−1a y,

where τax means the parallel translation from a to x along the geodesic arc

⌣ax and

Expa is the exponential map at a ∈ M . In this way we get a loop 〈M, ·a, a〉 with

neutral element a ∈ M . This construction justified the role of smooth quasigroupsand loops in differential geometry. But at that time the fundamental question,when the system of smooth loops on a manifold is a system of geodesic loops forsome affinely connected space, was still open.

4. Later, under the influence of [O. Loos 69], Kikkawa introduced the notion ofhomogeneous Lie loops and developed their theory. Actually, such loops are leftA-loops , that is, loops with the identity

(8)

ℓ(a , b) ◦ Lx ◦ [ℓ(a , b)]−1 = Lℓ (a , b) x ,

L c zdef= c · z , ℓ(a, b)

def= (La·b)

−1 ◦ La ◦ Lb .

380 L.V. Sabinin

This means that ℓ(a, b) is an automorphism of the considered loop. He introducedin this case the canonical reductive affine connection and the proper infinitesimalobject, a triple Lie algebra [K. Yamaguti 58a, b] (a binary-ternary linear algebrawith identities). See [M. Kikkawa 72, 73, 74, 75a, b, c, 84 a, b, 85, 91].

5. In 1972 Sabinin [L.V. Sabinin 72a, b, c] established the fundamental relationsbetween homogeneous spaces and left loops. As it is known, any left homogeneousspace G/H with a fixed cross section Q defines a left loop on itself by projectingon Q the group product of two elements from Q along a left coset:

(9) ∀ q1, q2 ∈ Q, q1 ∗ q2def= πQ(q1 · q2)

(this is due to [R. Baer 39, 40]). But there is a converse construction discoveredby Sabinin [L.V. Sabinin 72a, b, c]: by means of a left loop and its transassociant ,one may uniquely construct a left homogeneous space (semidirect product of aloop by its transassociant) and its cross-section such that the left loop induced onit by projecting along the left cosets is isomorphic to the original left loop.This result was generalized to left quasigroups, as well.Note that despite the fact that the above results are purely algebraic, they

are most important in the smooth setting (after obvious localization). Thus theBaer-Sabinin construction was established.

6. In [L.V. Sabinin, L.B. Sharma 76] the notion of Bol loop was generalized toso-called left half Bol loops and corresponding examples were given. This concepthas played an important role in current research. See [L.V. Sabinin, L.V. Sbitneva94], [L.V. Sabinin, Yu.A. Selivanov 94]. For the smooth case, see [L.V. Sabinin,C. Castillo 99].

7. In [L.V. Sabinin 77], the fundamental concepts of odule and odular struc-ture were introduced and applied to affinely connected manifolds. The notionof geodesic loop (Kikkawa) was enriched by the introduction of the canonicalunary operations (which resulted in the notion of geodesic odules and diodules).In this way a purely algebraic description of an affinely connected space as anodular (diodular) universal algebra with the geoodular identities was obtained.Such an algebraic description is impossible in the language of loops only. Inthis scheme an affinely connected manifold (M,∇) is a smooth universal algebra〈M, L, N, (ωt)t∈R〉 with ternary operations

(10) L(x, a, y) = Laxy = x ·

ay, N(x, a, y) = Na

xy = x+a

y

and binary operations

(11) ωt(a, x) = tax


such that, for any fixed a ∈ M , 〈M, ·a, a , (ta)t∈R〉 is a smooth local left odule,

which means that 〈M, ·a, a〉 is a loop with two-sided neutral a ∈ M and

(t+ u)ax = tax ·a

uax (monoassociativity),(12)

ta(uax) = (tu)ax (pseudoassociativity),(13)

1ax = x (unitarity).(14)

2) 〈M,+a, a, (ta)t∈R〉 is a local vector space.

3) The identities

Lu z xt z x ◦ Lz

t z x = Lzu z x (the first geoodular identity),(15)

Lzx ◦ t z = t x ◦ Lz

x (the second geoodular identity),(16)

Lzx (y +

zw) = Lz

x y +x

Lzx w , (the third geoodular identity).(17)

are valid.Such an algebra is called a geodiodular algebra. A geodiodular algebra has a

natural affine connection defined by

(18)∇X a

Y =

{

d

dt( [ (La

g (t) )∗, a ]−1 Yg (t) )

}

t=0,

g (0) = a , g (0) = Xa ,

Y being a vector field, Lacq = L(c, a, q) (see (10)).

The initial geodiodular structure may be restored by means of its natural con-nection ∇ as

L (x, a, y) = x ·a

y = Expx τax Exp

−1a y ,(19)

ω t (a , z) = t a z = Expa t Exp−1a z ,(20)

N(x, a, y) = x +a

y = Expa ( Exp−1a x + Exp−1a y) .(21)

Later, an algebraic study of affine connections was presented in the Ph.D. Disser-tation of our student Afanasiev [A.J. Afanasiev 84].In addition, some important classes of affinely connected spaces (reductive,

symmetric, etc.) were described by algebraic identities. Thus the identities

(22)

{

Lba L(x, y, z) = L (Lb

a x, Lba y, Lb

a z) ,

Lba ωt(x, y) = ωt (L

ba x, Lb

a y)

382 L.V. Sabinin

describe reductive spaces , and the identities

(23)

{

(−1)a L(x, y, z) = L((−1)ax, (−1)ay, (−1)az),

(−1)a ωt(x, y) = ωt((−1)ax, (−1)ay).

describe symmetric spaces . On this matter, see [L.V. Sabinin 81, 91b, 99a, b].In the Dr.Ph. Dissertation [H.M. Karanda 72], the fundamental fact that a

geodesic loop of symmetric space is a left Bol loop was disclosed. Later, it wasshown [L.V. Sabinin 81] that a locally symmetric space is nothing but a C3-smoothleft Bol-Bruck loop, that is a left Bol loop with the left Bruck identity

(24) (x · y) · (x · y) = x · (y · (y · x)).

Equivalently, one can replace (24) by the automorphic inverse identity

(x · y)−1 = x−1 · y−1.

It is interesting that in 1977 we did not know the results of Kikkawa[M. Kikkawa 64].The above results have introduced a new ideology into geometry which allows

one to treat an affinely connected space as a smooth algebraic system (nonlineargeometric algebra and nonassociative geometry).

8. Despite the work of A. Mal’cev and Kikkawa, it still was valuable to explorecertain classes of smooth loops close to Lie groups. Since a geodesic odule of anysymmetric space turned out to be a Bol loop, it was significant to study smoothBol loops.The infinitesimal theory of C3-smooth left (right) Bol loops was developed by

L. Sabinin and his student P. Miheev [L.V. Sabinin, P.O. Miheev 82b, 84, 85a, b],[P.O. Miheev 86a]. For a contemporary treatment, see [L.V. Sabinin 99b].In the above works the proper infinitesimal object, a binary-ternary linear

algebra with the identities

(25)

{

(η; ξ, ξ) = 0, (ξ; η, ζ) + (η; ζ, ξ) + (ζ; ξ, η) = 0,

((ξ; ν, τ); η, ζ) + (ξ; (η; ν, τ), ζ) + (ξ; η, (ζ; ν, τ)) = ((ξ; η, ζ); ν, τ)

(triple Lie algebra) and

(26)ξ · ξ = 0 ,

((τ · ζ); ξ, η) = (τ ; ξ, η) · ζ + τ · (ζ; ξ, η) + ((ξ · η); τ, ζ) + (τ · ζ) · (ξ · η) ,

was introduced. Such an object is called a Bol algebra.If 〈Q, ·, ε〉 is a C3-smooth left Bol loop and

(27) Aλ(x) =

[

∂(x · y)α

∂yλ

]

y=ε

∂

∂xα


are so-called left fundamental vector fields then

(28)(ξ · η) = [Aλ, Aµ](ε)ξ

ληµ,

(ξ; η, ζ) = [Aλ, [Aµ, Aν ]](ε)ξληµζν

define the tangent Bol algebra on V = Tε(Q).A Bol algebra uniquely defines a local left Bol loop and vice versa, in full

analogy with the Lie algebras–Lie groups theory. It is interesting that initiallythis theory was obtained by means of differential-geometric machinery, but later(see [L.V. Sabinin 91b, 99b]), it was elaborated in a more algebraic way.

9. The first comprehensive account of the geometric smooth quasigroups andloops theory was given in [L.V. Sabinin 81]. For a contemporary treatment of thewhole theory, see [L.V. Sabinin 99b].

10. Along with the above, geometric studies by means of the smooth quasigroupand loop approach took place. Generalized symmetric spaces were treated in thequasigroup language [L.V. Sbitneva 79, 82a, 82c, 84a, b]. This resulted in theconcept of a perfect s-space, which was well studied in algebraic and geometricways.

11. In 1988 the idea to create a general infinitesimal theory of smooth loops wascompletely realized in [L.V. Sabinin 88b]. The infinitesimal object in this theoryis much more complicated (a so-called hyperalgebra).

Thus, let 〈Q, · , ε〉 be a local Ck-smooth (k ≥ 3) loop, let

Aij (x) =

[

∂ (x · y )i

∂yj

]

y=ε

be its left fundamental vector fields (see (27)), and let Exp be the exponentialmap.We recall that, by definition,

(29)d ( Exp tq )i

dt= Ai

j ( Exp tq ) qj , Exp (0) = ε ,

and adjoin to the loop its canonical operations

(30) t x = Exp tExp−1 x , x+ y = Exp (Exp−1 x+ Exp−1 y ).

Let us also consider l (a, b) = [ ℓ (a, b) ]∗, ε and C ij k(x) defined by the unique

representation

(31) [Aj , Ak ] (x) = C ij k (x)Ai (x) .

384 L.V. Sabinin

We introduce the functions

(32)cijk (q) = C i

j k ( Exp q ) ,

l pm (v , w) = l p

m ( Exp v , Expw ) ; q, v, w ∈ T ε (Q) .

(Note that in the normal coordinates Exp q = q .)Let us define on T ε (Q) = V the operations

(33)ν (v, w) = l p

m (v, w)wm ( ∂ p ) ε ,

d (q, w) = cijk (q) q

jw k ( ∂ i ) ε .

The tangent vector space T ε (Q) = V equipped with these operations is called theν-hyperalgebra tangent to the loop 〈Q, · , ε 〉. Such an algebra possesses propertieswhich we collect in the following definition.

Definition. Let V be a vector space (over an arbitrary field K ), dimV = n,d (q, w) be a binary operation on V admitting a representation in an arbitrarybasis e1, . . . , en of the form

d (q, v) = dkij (q) q

i vj ek

and such that d (q, q) = 0. Then we say that V is a hyperalgebra.If, additionally, a binary operation ν (v, w) is given on V , and ν (v, w) admits

the representation in an arbitrary base e1, . . . , en of the form

ν (q, w) = νij (q, w)w

j ei ,

where ν (0, w) = w, νij (v, 0)wj = wi, then we say that V is a ν-hyperalgebra

(with multioperator ν).

Proposition. Any Cr-smooth (r ≥ 1) ν-hyperalgebra determines uniquely alocal Cr-smooth loop 〈Q, · , ε 〉 such that its tangent ν-hyperalgebra is isomor-phic to the initial ν-hyperalgebra.Morphisms of smooth loops induce morphisms of the corresponding ν-hyper-

algebras and vice versa.

Although the operations ν and d are uniquely defined, the above indicatedrepresentations for them are not unique. If ν and d are analytic then we canobtain a countable system of multilinear operations (with identities) which isequivalent to the original ν-hyperalgebra (under some conditions of convergence).We call such a system of multilinear operations also a ν-hyperalgebra.

A Ck-smooth (k ≥ 2) loop is right monoalternative, that is, (x · ty) · uy =x · (t+ u)y (t, u ∈ R), if and only if for its tangent ν-hyperalgebra ν (v, w) = w.

In different applications one needs derivatives of ν and d . For example, it isthe case for geometric odules, where one needs first derivatives of ν. This leads


us to a modification of the concept of ν-hyperalgebra (equivalent to the initialone). In such a way an F -hyperalgebra can be constructed.Let us note, finally, where, in the context of ν-hyperalgebra, Lie groups appear.

A C3-smooth loop is a Lie group if d (v, w) is bilinear and satisfies the identi-ties of a Lie algebra, and ν (v, w) = w (i.e., the loop is right monoalternative).Additional identities in a loop influence very much the structure of its tangentν-hyperalgebra (the identity of associativity demonstrates such a case).

Comment. Roughly speaking, in the analytic case one needs a countable set ofmultilinear operations instead of one binary operation as in the case of Lie groups— Lie algebras, or binary and ternary operations as in the case of Bol loops —Bol algebras. Of course, some (rather weak) identities should be added.

The complete treatment of the infinitesimal theory of smooth loops may befound in [L.V. Sabinin 91b, 99b]. In this connection we note that the earliertreatment of such a theory [L.V. Sabinin, P.O. Miheev 86, 87, 90] (presented ina differential-geometric way) is now only of historical interest due to the unsatis-factory definition of hyperalgebra there.

12. The concept of odular structure (L.V. Sabinin) has given a rise, to a gener-alization of G-spaces [H. Buseman 55]. Thus in [O.A. Matveev 86, 87] the notionof geodetic space was introduced. This is a smooth manifold with a system ofsmooth unary operations (local or global) and characteristic identities

(34)ux(txy) = (ut)xy, 1xy = y, txy = (1− t)yx

(t, u ∈ R, x, y ∈ M).

We note here that any affinely connected space is a geodetic space. In the smoothcase any geodetic space can be equipped with a unique affine connection of zerotorsion with the same geodesics {txy}t∈R. For that it is enough to take the tangentaffine connection to the loopuscular structure L(x, z, y) = Lz

x y = (2)z(1/2)x y,

(35)∇X a

Y =

{

d

dt( [ (La

g (t) )∗, a ]−1 Yg (t) )

}

t=0,

g (0) = a , g (0) = Xa ,

Y being a vector field.O.A. Matveev developed the algebraic theory of geodesic maps for affinely

connected spaces (or, equivalently, for smooth geoodular spaces).

13. The problem of when a smooth odule is a geodesic odule for some affineconnection was solved in [L.V. Sabinin 87]. It was proved that a smooth odule〈Q, ·, ε, (t)t∈R〉 is geodesic for some affine connection if and only if it is geometric,that is if the identity of geometricity

(36)ℓ(x, y) ty = t ℓ(x, y) y ,

ℓ(x, y) = (Lx·y)−1 ◦ Lx ◦ Ly (t ∈ R, x, y ∈ Q),

386 L.V. Sabinin

is valid. The above affine connection is not unique and the arbitrariness of itschoice was described as well. On this matter, see also [L.V. Sabinin 95b].In this setting, the concept of holonomial odule was introduced [L.V. Sabinin

87] as an odule 〈Q, ·, ε (t)t∈R〉 with an additional ternary operation h(a, b;x) andthe system of identities

h (a , b ; t x) = t h (a , b ;x) (homogeneity identity),(37)

h (a , a · b ; t b) = ℓ (a , b) t b (joint identity),(38)

h (c · ta , c · ua ;h (c , c · ta ;x)) = h(c , c · ua ;x) (h-identity),(39)

h (ε , q ;x) = x (ε-identity).(40)

This structure uniquely describes the geoodular space by

(41) x ·a

y = x · h (a , x ; a\y) , t a y = a · t (a\y) (a\y = (La)−1y).

This notion is equivalent to the notion of geoodular structure in the sense thatit allows one to concentrate all information about a geoodular structure (locally)into one fixed geodesic odule [L.V. Sabinin 87, 91b, 99b]. One may also say thatthe above construction is, in particular, an algebraic version of the equations ofE. Cartan in polar coordinates for an affine connection.

14. A delicate analysis of Bol loops and left A-loops has given rise to the notionof hyporeductive and pseudoreductive loops , as well as to the notion of hyporeduc-tive homogeneous space (a generalization of the reductive space [P.K. Rashevski50, 51, 52], [S. Kobayashi, K. Nomizu 63, 69]). The proper infinitesimal the-ory has been constructed, see [L.V. Sabinin 90b, 90c, 90d, 91a, 99b]. Later adifferential-geometric treatment of the above was given [I.A. Nourou 92]. Onsome generalizations, see [N.A. Gluhova 91].

15. The very interesting problem of describing spaces of constant curvature (and,more generally, projectively flat spaces) in a purely algebraic way was solved in[L.V. Sabinin, O.A. Matveev, S.S. Yantranova 86], [L.V. Sabinin, S.S. Yantranova84], [S.S. Yantranova 89]. It turns out that a geodesic diodule Q of such a spaceis a left Bol-Bruck loop with the pseudolinear identity

(42)x · y = α(x, y)x + β(x, y)y,

x, y ∈ Q; α(x, y), β(x, y) ∈ R.

This result was also generalized to symmetric spaces of index 1.

16. Meanwhile, analyzing the quasigroup structure of generalized symmetricspaces, L.V. Sabinin and L.L. Sabinina discovered the remarkable transsymmetricspaces [L.V. Sabinin, L.L. Sabinina 90, 91]. In the quasigroup language, these aresimply smooth left quasigroups satisfying the left F -identity

(43) x · (y · z) = (x · y) · (Fx · z) (F : Q → Q)


with the property of correctness: there exist (ρFx)−1, where ρz x

def= x · z (not

everywhere defined right division).The comprehensive theory of transsymmetric spaces in the language of smooth

left F -quasigroups was developed in [L.L. Sabinina 92]. See, also, [L.L. Sabinina93, 94, 95], [L.V. Sabinin, L.L. Sabinina 95]. Perfect transsymmetric spaces wereintroduced and studied in [L.V. Sabinin, L.L. Sabinina, L.V. Sbitneva 99].

17. Some first steps to solve the generalized fifth problem of Hilbert for loopswere undertaken in [L.V. Sabinin, L.L. Sabinina, R. Jimenez 97]. In particular, itwould be interesting to know, when a topological left Bol loop is analytic?

18. At this time the problem of constructing a theory of smooth loop actions(which would be helpful in applications to physics) is very significant. Instead ofa group 〈Q, ·, ε〉 action,

a ∈ Q 7→ (fa :M → M), fa ◦ fb = fa·b, fε = id,

one may try, for example,

(44) a ∈ Q 7→ (fa :M → M), fafbx = fa

x

· bx, fε = id, a, b,∈ Q,

where {〈Q,x· , ε〉}x∈Q is a family of loops. There is some preliminary non-rigorous

treatment of the matter in [I.A. Batalin 81]. As to Bol loops actions , the work[T. Nono 61] may give some hint.

19. Finally we would like to indicate some results on applications of smooth quasi-groups and loops. See [M.V. Karasev, V.P. Maslov 91] (Nonlinear Poisson brack-ets.), [P. Kuusk, J. Ord, E. Paal 94, 95] (General relativity), [J. Lohmus, E. Paal,L. Sorgsepp 94] (Survey on non-associative methods in physics), [A.I. Nesterov 89,90, 97] (Non-associative methods in physics), [E.P. Osipov 89] (Anomalies in fieldtheory), [E. Paal 88, 89] (Moufang symmetries in physics), [L.V. Sabinin, P.O. Mi-heev 93] (Special relativity), [L.V. Sabinin, A.I. Nesterov 97a] (Thomas preces-sion), [L.V. Sabinin, A.I. Nesterov 97b] (Generalized coherent states), [A. Un-gar 90, 94, 97] (Special relativity), [D.V. Yuriev 87, 92] (Chiral anomalies; Non-commutative geometry), [M. Bangoura 94] (Poisson brackets in mechanics).

References

I.L. Afanasiev[84] Algebraic structures of an affine connection, (Russian), Ph.D. Dissertation, Friendship

of Nations University, Moscow, 1984.

M.A. Akivis[78] Geodesic loops and local triple systems in a space with an affine connection, (Russian),

Siberian Math. J. 19 (1978), no. 2, 243–253; English transl.: Siberian Math. J. 19 (1978),no. 2, 171–178, MR 58 (1979) #7438.

388 L.V. Sabinin

M.Yu. Al-Houjeiri[87] Bol algebras for involutive pair of rank 1, (Russian), Ph.D. Dissertation, Inst. of Math.

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Department of Mathematics, Quintana Roo University, 77019 Chetumal, Quintana

Roo, Mexico

E-mail : [email protected]

(Received October 7, 1999)

smooth quasigroups and loops: forty-ﬁve years of...

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